Due to untrustworthy network, Alice could miss signed message either purposely or unwillingly (under manipulation), but what matters is that only when Alice receive a valid signature with updated states/balances (that she could compared with her locally stored last state), only then will she conceive the payment as secure and offer service to Bob/sender. In case of interrupted/corrupted message delivery, Alice could ask Bob to signed again with the same “updated states”. No one can double-spend the money.

In terms of http connection, the “payment” is just a signed message, shouldn’t be a problem if they have a smartphone and a normal messaging app for simplicity~